Theoretical Investigation of HER and OER Electrocatalysts Based on the 2D R-graphyne Completely Composed of Anti-Aromatic Carbon Rings

Based on the DFT calculations, two-dimensional (2D) R-graphyne has been demonstrated to have high stability and good conductivity, which can be conducive to the relevant electrocatalytic activity of the material. Different from the poor graphene, R-graphyne, which is completely composed of anti-aromatic structural units, can exhibit certain HER catalytic activity. In addition, doping the TM atoms in Group VIIIB can be considered an effective strategy to enhance the HER catalytic activity of R-graphyne. Particularly, Fe@R-graphyne, Os@R-graphyne, Rh@R-graphyne and Ir@R-graphyne can exhibit higher HER catalytic activities due to the formation of more active sites. Usually, the shorter the distance between the TM and C atoms is, the better the HER activity of the C-site is. Furthermore, doping Ni and Rh atoms of Group VIIIB can significantly improve the OER catalytic performance of R-graphyne. It can be found that ΔGO* can be used as a good descriptor for the OER activities of TM@R-graphyne systems. Both Rh@R-graphyne and Ni@R-graphyne systems can exhibit bifunctional electrocatalytic activities for HER/OER. In addition, all the relevant catalytic mechanisms are analyzed in detail. This work not only provides nonprecious and highly efficient HER/OER electrocatalysts, but also provides new ideas for the design of carbon-based electrocatalysts.


Introduction
Electrocatalytic water splitting for hydrogen production has great prospects in solving the global energy crisis caused by energy depletion and environmental pollution [1][2][3]. Electrolytic water splitting includes two half reactions: hydrogen evolution reaction (HER) and oxygen evolution reaction (OER). Owing to sluggish kinetics, electrocatalysts are required to reduce the overpotentials of these two reactions and improve the reaction kinetics. In previous reports, some HER or OER electrocatalysts have been designed and developed [4][5][6][7][8][9]. So far, the benchmark electrocatalysts for HER and OER are Pt [10] and IrO 2 /RuO 2 [11,12], respectively. However, the scarcity and high cost of these precious metals have greatly hindered their large-scale applications. Therefore, many efforts have been made to design highly efficient HER/OER electrocatalysts by minimizing the use of precious metals.
At present, two-dimensional (2D) materials have attracted great attention from researchers because their unique planar structure with atomic thickness can bring some obvious advantages to catalytic reactions, such as providing a larger specific surface area and endowing them with abundant exposed active sites, allowing them to easily combine primitive unit cell are a = b = 6.011 Å . There are three nonequivalent C-C bonds, labeled as d1, d2 and d3 ( Figure S1), with the bond lengths of 1.242, 1.351 and 1.463 Å , respectively, which is consistent with the previous work [42].
In this study, we performed an ab initio molecular dynamics (AIMD) simulation [44] at 500 K with a time constant of 5000 fs to evaluate the thermal stability of R-graphyne. It is clear that there is a small energy fluctuation, and the structural integrity of the original atomic configuration can be maintained after a time interval of 5 ps (Figure 1b), indicating the good thermal stability of R-graphyne. Furthermore, we calculated the electron localization function (ELF) [45] to illustrate the type of chemical bonding between two adjacent C atoms. As shown in Figure 1c, the red regions of the ELF diagram demonstrate the formation of strong covalent bonds and highlight the stability of R-graphyne. In addition, we investigated the electronic property of R-graphyne by calculating the density of states (DOS). As shown in Figure 1d, the metallic behavior can be observed in R-graphyne, where the correlative states cross the Fermi level, indicating good conductivity. It is well known that good conductivity can be beneficial to the relevant electrocatalytic activity of the material. Subsequently, we investigated the HER electrocatalytic performance of R-graphyne by calculating the free energy of hydrogen adsorption (ΔGH*). For comparison, the well-known 2D graphene was also considered. It has been well proven that the catalytic activity of the HER closely correlates with the adsorption energy of a single hydrogen Figure 1. (a) The optimized structure of 2D R-graphyne and the obtained adsorption sites of H*, (b) the total energy curve of ab initio molecular dynamics (AIMD) simulation for R-graphyne at 500 K, where the inset is the configuration after 5000 fs, (c) the calculated electron location function (ELF) of the R-graphyne, (d) the density of states (DOS) of R-graphyne, where the Fermi level is set as 0, and (e) the calculated ∆G H* values for the different adsorption sites of R-graphyne and graphene at equilibrium potential.
In this study, we performed an ab initio molecular dynamics (AIMD) simulation [44] at 500 K with a time constant of 5000 fs to evaluate the thermal stability of R-graphyne. It is clear that there is a small energy fluctuation, and the structural integrity of the original atomic configuration can be maintained after a time interval of 5 ps (Figure 1b), indicating the good thermal stability of R-graphyne. Furthermore, we calculated the electron localization function (ELF) [45] to illustrate the type of chemical bonding between two adjacent C atoms. As shown in Figure 1c, the red regions of the ELF diagram demonstrate the formation of strong covalent bonds and highlight the stability of R-graphyne. In addition, we investigated the electronic property of R-graphyne by calculating the density of states (DOS). As shown in Figure 1d, the metallic behavior can be observed in R-graphyne, where the correlative states cross the Fermi level, indicating good conductivity. It is well known that good conductivity can be beneficial to the relevant electrocatalytic activity of the material.
Subsequently, we investigated the HER electrocatalytic performance of R-graphyne by calculating the free energy of hydrogen adsorption (∆G H* ). For comparison, the wellknown 2D graphene was also considered. It has been well proven that the catalytic activity of the HER closely correlates with the adsorption energy of a single hydrogen atom on the material's surface [46]. Therefore, ∆G H* can be identified as a representative indicator to measure HER activity [13,18,23,30,47]. Usually, the smaller absolute value of ∆G H* means a better HER catalytic activity of the material [46]. The calculation formula of ∆G H* is provided in the section on computational methods.
As shown in Figure 1e, two different adsorption sites (namely, T C1 and T C2 ) can be obtained on the 2D R-graphyne, and the calculated ∆G H* values are 0.450 eV and 0.423 eV, respectively, indicating a certain catalytic activity of the HER. Obviously, compared with the inert HER activity of graphene (1.642 eV), the R-graphyne can show a good trend of change in HER catalytic performance, although both of them can be regarded as 2D π-conjugated structures composed of sp 2 -or sp-hybridized C atoms. This can be mainly due to the different structural units they are composed of. Specifically, graphene consists of aromatic six-membered carbon rings meeting the (4n + 2) rule of π electrons, which can be reflected well by the relevant molecular orbitals ( Figure 2). Comparatively, R-graphyne is composed of four-membered and sixteen-membered carbon rings with anti-aromatic characteristics satisfying the 4n rule, which can also be reflected by the molecular orbitals of the related carbon rings ( Figure 2). Furthermore, we have also carried out the NICS calculations, which can usually be used as a reliable indicator to determine the aromaticity [48]. Indeed, the aromaticity of six-membered carbon rings (−7.97/−10.11) and the anti-aromaticity of four-membered (27.85/18.78) or sixteen-membered (15.65/13.85) carbon rings can be further supported by the calculated negative and positive NICS(0)/NICS(1) values, respectively ( Figure 2). Clearly, compared with the aromatic case of graphene, the presence of anti-aromaticity can bring more active C atoms, which leads to a certain HER catalytic performance of R-graphyne. Designing the 2D carbon allotrope completely composed of anti-aromatic structural units can be considered as an effective strategy to boost the HER catalytic activity of related carbon materials. atom on the material's surface [46]. Therefore, ΔGH* can be identified as a representative indicator to measure HER activity [13,18,23,30,47]. Usually, the smaller absolute value of ΔGH* means a better HER catalytic activity of the material [46]. The calculation formula of ΔGH* is provided in the section on computational methods. As shown in Figure 1e, two different adsorption sites (namely, TC1 and TC2) can be obtained on the 2D R-graphyne, and the calculated ΔGH* values are 0.450 eV and 0.423 eV, respectively, indicating a certain catalytic activity of the HER. Obviously, compared with the inert HER activity of graphene (1.642 eV), the R-graphyne can show a good trend of change in HER catalytic performance, although both of them can be regarded as 2D π-conjugated structures composed of sp 2 -or sp-hybridized C atoms. This can be mainly due to the different structural units they are composed of. Specifically, graphene consists of aromatic six-membered carbon rings meeting the (4n + 2) rule of electrons, which can be reflected well by the relevant molecular orbitals ( Figure 2). Comparatively, R-graphyne is composed of four-membered and sixteen-membered carbon rings with anti-aromatic characteristics satisfying the 4n rule, which can also be reflected by the molecular orbitals of the related carbon rings ( Figure 2). Furthermore, we have also carried out the NICS calculations, which can usually be used as a reliable indicator to determine the aromaticity [48]. Indeed, the aromaticity of six-membered carbon rings (−7.97/−10.11) and the anti-aromaticity of four-membered (27.85/18.78) or sixteen-membered (15.65/13.85) carbon rings can be further supported by the calculated negative and positive NICS(0)/NICS(1) values, respectively ( Figure 2). Clearly, compared with the aromatic case of graphene, the presence of anti-aromaticity can bring more active C atoms, which leads to a certain HER catalytic performance of R-graphyne. Designing the 2D carbon allotrope completely composed of anti-aromatic structural units can be considered as an effective strategy to boost the HER catalytic activity of related carbon materials.   From the above discussions, we found that the pristine R-graphyne can exhibit certain HER catalytic activity, which is mainly due to the anti-aromaticity of its structural unit rings. However, based on the fact that the calculated ∆G H* values are not close to zero, we propose a promising approach through embedding the Group VIIIB elements (i.e., Fe, Co, Ni, Ru, Rh, Pd, Os, Ir and Pt) into R-graphyne with its unique structure to construct TM@R-graphyne to effectively improve the performance of the HER.
As shown in Figure S1, two possible doping sites of transition metal (TM) atoms are considered, that is, the TM atom is connected with four C atoms at the corner of the sixteen-membered ring (S1) and the TM is located over the top of the four-membered ring (S2). The calculated results show that configuration S1 can be much lower in energy than configuration S2, indicating the higher structural stability. Therefore, doping configuration S1 is considered in this study. It can be found that the planar configuration of R-graphyne can be well maintained after embedding the TM atoms of Group VIIIB ( Figure S2). The calculated lattice parameters of TM@R-graphyne can be in the range of a = b = 11.934~11.994 Å (Table S1), which are comparable to the corresponding 2 × 2 supercell unit of the pristine R-graphyne. In addition, the calculated binding energies of TM-doped R-graphyne systems can be largely negative in the range of −2.509~−4.773 eV (Table S1), indicating that these VIIIB TM atoms can be stably anchored in R-graphyne. We have also performed the AIMD calculations for the TM@R-graphyne structure at 500 K for 5 ps ( Figure S3). It can be found that the energy of TM@R-graphyne oscillates only in a small range during the AIMD simulation and the geometric structure can be basically maintained, indicating high thermal stability. Moreover, we investigate the electronic property of TM@R-graphyne by calculating the corresponding density of states (DOS). As shown in Figure S4, the metallic behavior can still be observed in these TM@R-graphyne systems, which suggests good conductivity. High stability and good conductivity can inspire us to explore the relevant electrocatalytic performance of TM@R-graphyne.
Here, we investigate the HER catalytic activity for a series of TM@R-graphyne systems (TM = Fe, Ru, Os, Co, Rh, Ir, Ni, Pd and Pt). A total of nine possible adsorption sites over C atoms and TM atom are considered, namely T C1~TC8 and T TM , respectively ( Figure 3a). Initially, we evaluate the HER catalytic performance of R-graphyne doped with Fe, Ru and Os atoms in the same column of Group VIIIB by calculating the ∆G H* values of relevant possible adsorption sites ( Figure 3b and Table S2). Our computed results reveal that when doping Fe atom into R-graphyne, almost all the C-sites (except for T C1 ) can exhibit much smaller ∆G H* values in the range of −0.140~0.250 eV. Specifically, the calculated ∆G H* values of T C2, T C3, T C4 , T C5 , T C6 , T C7 and T C8 sites on Fe@R-graphyne can be as small as 0.239, −0.140, 0.122, 0.181, 0.235, 0.250 and 0.237 eV, respectively, uniformly suggesting considerably high HER catalytic activity. Moreover, the T Fe site can also exhibit considerably high HER activity, in view of the small ∆G H* value of 0.142 eV. Clearly, Fe@R-graphyne can possess excellent HER catalytic activity, much higher than the pristine R-graphyne, due to the appearance of more highly active sites with much smaller ∆G H* values.
When doping Ru and Os atoms in the higher period, high HER activity can also be observed, although the number of highly active sites on Ru@R-graphyne and Os@Rgraphyne decreases slightly compared with Fe@R-graphyne. Specifically, doping Ru can make the ∆G H* values of some C-sites become smaller, including T C2 (−0.104), T C3 (−0.014), T C4 (0.316) and T C5 (0.328 eV), all of which can possess higher HER activity than the pristine R-graphyne. In particular, some C atoms around the Ru-doped site, including T C2 (−0.104 eV) and T C3 (−0.014 eV), can exhibit near-zero ∆G H* values, suggesting their considerably high HER catalytic activities. In contrast, the absolute ∆G H* values of the remaining C-sites (except for T C1 ) are in the range of 0.425~0.503 eV, all of which are comparable to the corresponding those of pristine R-graphyne, implying that these C-sites can maintain the original HER activity. Similarly, the doping of Os can also bring the smaller ∆G H* values to most of the C-sites, including T C1 (−0.140 eV), T C2 (0.272 eV), T C3 (−0.263 eV), T C4 (0.196 eV) and T C5 (0.334 eV), suggesting their better HER activities than the corresponding ones on the pristine R-graphyne. Comparatively, the calculated ∆G H* values of the remaining C-sites on Os@R-graphyne are in the range of 0.461~0.496 eV, indicating that these C-sites can maintain the original HER activity. It can be found that for Ru@R-graphyne and Os@R-graphyne, the closer distances between the relevant C atom and the TM-doping site can generally bring a better HER catalytic activity of the C-site, mainly because it can be activated more effectively. In addition, with the increase in the periodic number of the doped TM atom, the adsorption strength of H* at the metal site will increase, and the corresponding HER catalytic activity will gradually decrease, as revealed by the calculated ∆G H* results of three TM sites including T Fe (0.142 eV), T Ru (−0.501 eV) and T Os (−0.778 eV). The calculated ΔGH* values for different adsorption sites on TM@R-graphyne at equilibrium potential. Note that the HER catalytic performance can be indicated by the distinct colors, where the darker red color means higher activity.
When doping Ru and Os atoms in the higher period, high HER activity can also be observed, although the number of highly active sites on Ru@R-graphyne and Os@R-graphyne decreases slightly compared with Fe@R-graphyne. Specifically, doping Ru can make the ΔGH* values of some C-sites become smaller, including TC2 (−0.104), TC3 (−0.014), TC4 (0.316) and TC5 (0.328 eV), all of which can possess higher HER activity than the pristine R-graphyne. In particular, some C atoms around the Ru-doped site, including TC2 (−0.104 eV) and TC3 (−0.014 eV), can exhibit near-zero ΔGH* values, suggesting their considerably high HER catalytic activities. In contrast, the absolute ΔGH* values of the remaining C-sites (except for TC1) are in the range of 0.425~0.503 eV, all of which are comparable to the corresponding those of pristine R-graphyne, implying that these C-sites can maintain the original HER activity. Similarly, the doping of Os can also bring the smaller ΔGH* values to most of the C-sites, including TC1 (−0.140 eV), TC2 (0.272 eV), TC3 (−0.263 eV), TC4 (0.196 eV) and TC5 (0.334 eV), suggesting their better HER activities than the corresponding ones on the pristine R-graphyne. Comparatively, the calculated ΔGH* values of the remaining C-sites on Os@R-graphyne are in the range of 0.461~0.496 eV, indicating that these C-sites can maintain the original HER activity. It can be found that for Ru@R-graphyne and Os@R-graphyne, the closer distances between the relevant C atom and the TM-doping site can generally bring a better HER catalytic activity of the C-site, mainly because it can be activated more effectively. In addition, with the increase in the periodic number of the doped TM atom, the adsorption strength of H* at the metal site will increase, and the corresponding HER catalytic activity will gradually decrease, as revealed by the calculated ΔGH* results of three TM sites including TFe (0.142 eV), TRu Overall, a series of TM@R-graphyne systems (TM = Fe, Ru and Os) can uniformly possess high HER catalytic activity, in view of the formation of highly active sites with much smaller ∆G H* values. In particular, doping Fe can induce higher HER catalytic activity due to the presence of more active sites.
Subsequently, we investigate the HER catalytic activity of the R-graphyne systems doped with Co, Rh and Ir atoms with more valence electrons in the same column of Group VIIIB by considering the relevant possible adsorption sites (Figure 3a,b). Our computed results show that for Co@R-graphyne, both the T C3 (0.068) and T Co (0.069 eV) sites can have the near-zero ∆G H* value, indicating the remarkable HER activity of these two sites. The absolute values of ∆G H* for other C-sites (except for T C1 ) are in the range of 0.369~0.449 eV, indicating that their HER activity can be superior to or comparable to the corresponding activity of pure R-graphyne. As for Rh@R-graphyne, most of the C-sites (including T C2, T C3 , T C4, T C5 and T C6 ) can also exhibit enhanced HER activity compared with the pristine R-graphyne, as reflected by their much smaller ∆G H* values in the range of −0.118~0.399 eV. Particularly, some C-sites near the Rh-doping site can even present very small ∆G H* values, namely T C2 (−0.118 eV) and T C3 (−0.037 eV), indicating the considerably high HER activity. In contrast, the absolute ∆G H* values of all C-sites on Ir@R-graphyne can be in the range of 0.055~0.449 eV, which are smaller than or close to the corresponding ones on the pure R-graphyne. Among them, the calculated ∆G H* values of carbon sites (including T C1 , T C2 , T C3 and T C4 ) near the Ir-doping site can be as small as −0.005, 0.246, −0.099 and 0.241 eV, respectively, reflecting considerably high HER activity. Similarly, with the increase inthe periodic number of TM atoms from Co to Ir, the HER catalytic activity of the TM site can present a gradually decreasing trend, which is reflected in the corresponding ∆G H* results of three TM sites including T Co (0.069 eV), T Rh (−0.497 eV) and T Ir (−0.581 eV).
Obviously, the HER activity on 2D R-graphyne can also be effectively improved by doping Co, Rh and Ir atoms with more valence electrons, and especially embedding Ir can more effectively enhance the HER activity due to the presence of more highly active sites. Similarly, it can also be observed that in this series of systems, the shorter the distance between the TM and C atoms, the higher the HER activity of the T C site.
Next, we continue to explore the HER catalytic performance of R-graphyne systems doped with Ni, Pd and Pt atoms in the same column, which can possess the most valence electrons in Group VIIIB. Our computed results show that due to the formation of active sites, doping Ni, Pd and Pt can improve the HER activity of R-graphyne, although their HER catalytic activity can be lower than that of R-graphyne doped with other TM atoms in Group VIIIB (TM = Fe, Ru, Os, Co, Rh and Ir). Specifically, the T C3 site (0.210 eV) on Ni@R-graphyne, the T C1 (0.240 eV) and T C3 (0.282 eV) sites on Pd@R-graphyne and the T C1 site (0.267 eV) on Pt@R-graphyne can exhibit relatively high HER activity. The calculated absolute ∆G H* values of the remaining C-sites on TM@R-graphyne can be in the range of 0.372~0.600 eV, 0.351~0.421 eV and 0.303~0.429 eV for TM = Ni, Pd and Pt, respectively, most of which can be comparable to the corresponding values of pristine R-graphyne, suggesting that these C-sites can basically maintain the original HER activity. In addition, all three relevant TM sites (Ni, Pd and Pt) can have largely positive ∆G H* values in the range of 0.586~0.887 eV (Table S2), indicating relatively poor HER activity.
Furthermore, we also discussed the effect of increasing the valence electron number of TM atoms on the HER catalytic activity of the doped R-graphyne systems. Specifically, with the increase in the valence electron number of 3d TM atoms (Fe, Co and Ni), the HER activity of TM@R-graphyne will reduce owing to the decrease in highly active sites. In contrast, the 4d Ru-and Rh-doped R-graphyne systems can have similar HER activity, which are better than the parallel Pd-doped R-graphyne. Similarly, 5d Os-and Ir-doped R-graphyne systems can also exhibit comparable HER catalytic activity, which are higher than the parallel Pt@Rgraphyne. Among them, Os@R-graphyne, Ir@R-graphyne, Rh@R-graphyne and especially Fe@R-graphyne can exhibit higher HER activity than the remaining TM@R-graphyne systems, due to the presence of more active sites.
Obviously, embedding the TM atoms in Group VIIIB can be considered an effective strategy to improve the HER catalytic activity of 2D R-graphyne completely composed of anti-aromatic structural units. The relevant C atoms near the TM-doping sites can be used as highly active sites in HER. Generally, shorter distances between the TM and C atoms can achieve better HER activity at the T C site. The enhancement of HER activity can be attributed to the fact that introducing the TM atoms of Group VIIIB can induce an electron transfer process between TM and its adjacent C atoms (Figure 4), which can effectively adjust the electron density on the relevant C atoms, thus endowing H* with an appropriate adsorption state.

Evaluation of OER Catalytic Activity of TM@R-graphyne
In addition to HER activity, we investigate the OER electrocatalytic performance of R-graphyne doped with TM atoms of Group VIIIB by calculating the Gibbs free energy difference of four elementary steps (∆G 1 , ∆G 2 , ∆G 3 and ∆G 4 ) to evaluate the overpotential of OER (η OER ). It is noteworthy that the adsorption of intermediates (*O, *OH and *OOH) at the TM site on TM@R-graphyne (TM = Fe, Os, Pd and Pt) can lead to severe structural deformation, thus they are excluded. Finally, five possible candidates can be obtained, including Ru@R-graphyne, Co@R-graphyne, Rh@R-graphyne, Ir@R-graphyne and Ni@Rgraphyne ( Figures 5 and S5).
anti-aromatic structural units. The relevant C atoms near the TM-doping sites can be used as highly active sites in HER. Generally, shorter distances between the TM and C atoms can achieve better HER activity at the TC site. The enhancement of HER activity can be attributed to the fact that introducing the TM atoms of Group VIIIB can induce an electron transfer process between TM and its adjacent C atoms (Figure 4), which can effectively adjust the electron density on the relevant C atoms, thus endowing H* with an appropriate adsorption state.

Evaluation of OER Catalytic Activity of TM@R-graphyne
In addition to HER activity, we investigate the OER electrocatalytic performance of R-graphyne doped with TM atoms of Group VIIIB by calculating the Gibbs free energy difference of four elementary steps (ΔG1, ΔG2, ΔG3 and ΔG4) to evaluate the overpotential of OER (ηOER). It is noteworthy that the adsorption of intermediates (*O, *OH and *OOH) at the TM site on TM@R-graphyne (TM = Fe, Os, Pd and Pt) can lead to severe structural deformation, thus they are excluded. Finally, five possible candidates can be obtained, including Ru@R-graphyne, Co@R-graphyne, Rh@R-graphyne, Ir@R-graphyne and Ni@R-graphyne ( Figures 5 and S5). For all four TM@R-graphyne systems (TM = Co, Ru, Rh and Ir), the third step (O*→*OOH) can be regarded as the potential-determining step (PDS) in the OER process (Figures 5a, c, and S5). Among them, the calculated ηOER values of Ru@R-graphyne (1.43 V) and Ir@R-graphyne (1.16 V) can be largely positive, indicating the inert OER catalytic activity. Comparatively, Co@R-graphyne (0.74 V) can have certain OER activity, and especially Rh@R-graphyne can exhibit considerably high OER performance considering the very low ηOER value of 0.48 V. As for Ni@R-graphyne, the second step (OH*→*O) can be considered as the PDS step of OER (Figure 5b), and the calculated ηOER value can be as small as 0.31 V, indicating excellent OER catalytic activity. It is worth mentioning that the overpotentials of Ni@R-graphyne (0.31 V) and Rh@R-graphyne (0.48 V) can be comparable to or even lower than those of the state-of-the-art IrO2 (0.56 V) [12] and other previously reported MoC2 (0.45 V) [16] and RuO2 (0.42 V) [49] systems. Obviously, embedding Ni and Rh atoms in Group VIIIB can significantly improve the OER catalytic performance of R-graphyne.
Generally, the relationship between the Gibbs free energies (ΔGOH*, ΔGO* and ΔGOOH*) of three intermediates can be used to obtain the appropriate descriptors and thus For all four TM@R-graphyne systems (TM = Co, Ru, Rh and Ir), the third step (O*→*OOH) can be regarded as the potential-determining step (PDS) in the OER process (Figures 5a,c, and S5). Among them, the calculated η OER values of Ru@R-graphyne (1.43 V) and Ir@R-graphyne (1.16 V) can be largely positive, indicating the inert OER catalytic activity. Comparatively, Co@R-graphyne (0.74 V) can have certain OER activity, and especially Rh@R-graphyne can exhibit considerably high OER performance considering the very low η OER value of 0.48 V. As for Ni@R-graphyne, the second step (OH*→*O) can be considered as the PDS step of OER (Figure 5b), and the calculated η OER value can be as small as 0.31 V, indicating excellent OER catalytic activity. It is worth mentioning that the overpotentials of Ni@R-graphyne (0.31 V) and Rh@R-graphyne (0.48 V) can be comparable to or even lower than those of the state-of-the-art IrO 2 (0.56 V) [12] and other previously reported MoC 2 (0.45 V) [16] and RuO 2 (0.42 V) [49] systems. Obviously, embedding Ni and Rh atoms in Group VIIIB can significantly improve the OER catalytic performance of R-graphyne.
Generally, the relationship between the Gibbs free energies (∆G OH* , ∆G O* and ∆G OOH* ) of three intermediates can be used to obtain the appropriate descriptors and thus reveal the OER trend of catalysts [50][51][52][53][54][55]. As shown in Figure 6a, ∆G OH* can be specified as a function of ∆G O* , which can be expressed as ∆G OH* = 0.39∆G O* − 0.10 eV. Clearly, ∆G OH* can exhibit a strong linear correlation with ∆G O* considering the high determination coefficient (R 2 ) of 0.92. Moreover, ∆G OOH* can also scale linearly with ∆G O* , which can be expressed as ∆G OOH* = 0.44∆G O* + 2.86 eV with an R 2 of 0.90 (Figure 6b). Obviously, ∆G O* can be adopted as an effective descriptor for OER performance, because ∆G OOH* and ∆G OH* show strong linear correlation with ∆G O* at the same time.
pecially Rh@R-graphyne can exhibit considerably high OER performance considering the very low ηOER value of 0.48 V. As for Ni@R-graphyne, the second step (OH*→*O) can be considered as the PDS step of OER (Figure 5b), and the calculated ηOER value can be as small as 0.31 V, indicating excellent OER catalytic activity. It is worth mentioning that the overpotentials of Ni@R-graphyne (0.31 V) and Rh@R-graphyne (0.48 V) can be comparable to or even lower than those of the state-of-the-art IrO2 (0.56 V) [12] and other previously reported MoC2 (0.45 V) [16] and RuO2 (0.42 V) [49] systems. Obviously, embedding Ni and Rh atoms in Group VIIIB can significantly improve the OER catalytic performance of R-graphyne.
Generally, the relationship between the Gibbs free energies (ΔGOH*, ΔGO* and ΔGOOH*) of three intermediates can be used to obtain the appropriate descriptors and thus reveal the OER trend of catalysts [50][51][52][53][54][55]. As shown in Figure 6a, ΔGOH* can be specified as a function of ΔGO*, which can be expressed as ΔGOH* = 0.39ΔGO* − 0.10 eV. Clearly, ΔGOH* can exhibit a strong linear correlation with ΔGO* considering the high determination coefficient (R 2 ) of 0.92. Moreover, ΔGOOH* can also scale linearly with ΔGO*, which can be expressed as ΔGOOH* = 0.44ΔGO* + 2.86 eV with an R 2 of 0.90 (Figure 6b). Obviously, ΔGO* can be adopted as an effective descriptor for OER performance, because ΔGOOH* and ΔGOH* show strong linear correlation with ΔGO* at the same time.  Subsequently, we have drawn the diagram describing the relationship between η OER and ∆G O* , as shown in Figure 6c. It can be found that along with the increase in ∆G O* value, the calculated η OER value can gradually decrease for these five TM@R-graphyne (TM = Ru, Co, Rh, Ir and Ni) systems, where doping Ni can bring the smallest η OER value (0.31 V). Further, a contour map reflecting the trend of OER activity is also provided, which is divided into four regions based on the potential-determining step (Figure 7). The pink region in Figure 7 predicts the limiting overpotentials η of the TM@R-graphyne series, which can be as small as 0.29 V. It is worth mentioning that the newly designed Rh@Rgraphyne (0.48 V) can be close to the limit value, and especially Ni@R-graphyne (0.31 V) can be very close to it, indicating considerably high OER catalytic performance.
Molecules 2023, 28, x FOR PEER REVIEW 10 of 16 Subsequently, we have drawn the diagram describing the relationship between ηOER and ΔGO*, as shown in Figure 6c. It can be found that along with the increase in ΔGO* value, the calculated ηOER value can gradually decrease for these five TM@R-graphyne (TM = Ru, Co, Rh, Ir and Ni) systems, where doping Ni can bring the smallest ηOER value (0.31 V). Further, a contour map reflecting the trend of OER activity is also provided, which is divided into four regions based on the potential-determining step (Figure 7). The pink region in Figure 7 predicts the limiting overpotentials η of the TM@R-graphyne series, which can be as small as 0.29 V. It is worth mentioning that the newly designed Rh@R-graphyne (0.48 V) can be close to the limit value, and especially Ni@R-graphyne (0.31 V) can be very close to it, indicating considerably high OER catalytic performance. From the above discussions, we can understand that the adsorption strength of O* plays a crucial role in determining the OER catalytic performance of TM@R-graphyne. In order to better understand that the doping of Ni and Rh atoms can bring the appropriate adsorption state of O*, we have conducted an in-depth bonding analysis on the intercation between O* and Ni/Rh center. For comparison, the TM@R-graphyne with From the above discussions, we can understand that the adsorption strength of O* plays a crucial role in determining the OER catalytic performance of TM@R-graphyne. In order to better understand that the doping of Ni and Rh atoms can bring the appropriate adsorption state of O*, we have conducted an in-depth bonding analysis on the intercation between O* and Ni/Rh center. For comparison, the TM@R-graphyne with poor catalytic activity in the OER is also considered by sampling Ru@R-graphyne. Figure 8 shows the partial density of states (PDOSs) of the p orbitals of O atoms and d orbitals of relevant TM atoms, where bond properties between O* and the metal site are presented by drawing different ranges of molecular orbitals. It can be found that when the doped Ru atom is used as an adsorption site, the center of overlapping O-p and Ru-d orbitals is in the π bonding region (Figure 8a), which can lead to a relatively strong interaction between O* and the TM site, thus resulting inthe poor OER activity. In contrast, when Rh atom is doped, the overlapping center of the O-p and Rh-d orbitals will move into the π* antibonding region ( Figure 8b). The existence of an antibonding characteristic will effectively weaken the interaction strength between O* and the TM site, which can result in a relatively sutiable adsorption state of O*, thus significantly improving the catalytic activity of the OER. Further, when Ni is introduced into R-graphyne, the overlapping center of the O-p and Ni-d orbitals in the π* antibonding region can continue to move towards the Fermi level (Figure 8c), indicating that the antibonding characteristics can be increased to some extent, which will bring a more appropriate adsorption state of O*, thus further enhancing the OER activity. In general, in addition to HER activity, the OER catalytic performance of 2D R-graphyne composed entirely of anti-aromatic structural units can also be effectively improved by doping the TM atoms in Group VIIIB. In particular, it is highly anticipated that both the Ni@R-graphyne and Rh@R-graphyne systems can be used as promising In general, in addition to HER activity, the OER catalytic performance of 2D Rgraphyne composed entirely of anti-aromatic structural units can also be effectively improved by doping the TM atoms in Group VIIIB. In particular, it is highly anticipated that both the Ni@R-graphyne and Rh@R-graphyne systems can be used as promising bifunctional electrocatalysts for HER and OER.

Computational Methods
The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof exchange correlation functional [56] is used to perform all the density functional theory (DFT) calculations of the studied systems within the framework of the Vienna ab initio simulation package (VASP) [57,58]. Spin-polarized calculations have been considered in this study. The cut-off energy for the plane-wave basis was set as 400 eV and a semiempirical van der Waals (vdW) correction proposed by Grimme (DFT-D2) was employed to account for the dispersion interactions [56,59]. The ion-electron interaction was described by the projected enhancement wave (PAW) potential [60]. The 3 × 3 × 1 k-points were used for the structural optimization and 84 k-points were used to calculate the density of states (DOSs). The convergence criteria of total energy and force were 10 −4 eV and 0.02 eV A −1 , respectively. A vacuum space of 20 Å was applied along the z-direction to avoid the interactions between adjacent images. In addition, the nucleus-independent chemical shift (NICS) [61,62] calculations were conducted at the B3LYP/6-31G(d) level using the Gaussian 09 program package [63] to evaluate the aromatic properties of the relevant carbon rings in the studied system.
In order to evaluate the stability of transition-metal-doped R-graphyne system, the binding energy (E b ) is calculated according to the following formula: where E TM@R-graphyne , E TM and E R-graphyne represent the total energy of TM@R-graphyne, single transition metal atom (TM) and the pristine R-graphyne, respectively. A more negative E b value can mean higher stability. The HER performance can be evaluated by calculating the free energy of the reaction (∆G H* ) according to the following formula: Here, ∆E H* represents the energy difference of H* adsorption. ∆ZPE and ∆S are the changes of zero-point energy and entropy of H* adsorption, respectively. In this study, T∆S and ∆ZPE were obtained by following the scheme proposed by Nørskov et al. [46]. Specifically, ∆S was calculated by using the equation ∆S = S(H*) − 1/2S(H 2 ) ≈−1/2S(H 2 ), because the vibrational entropy of H* can be very small and negligible. Considering that TS (H 2 ) is 0.410 eV for H 2 at 298 K and 1 atm, the corresponding T∆S was determined as −0.205 eV. In addition, the equation ∆ZPE = ZPE(H*) − 1/2ZPE(H 2 ) was applied to estimate ∆ZPE for H*. It is worth mentioning that our computed ZPE(H 2 ) value is about 0.295 eV, which is close to the value reported by Nørskov et al. [46].
The OER process is usually carried out through the four-electron transfer steps, which can be summarized as follows: The Gibbs free energy change (∆G) of each elemental step can be obtained by the following expression: where ∆E is the adsorption energy for the relevant intermediates involving OH, O and OOH. ∆ZPE and ∆S are the zero-point energy change and the change in entropy, respectively. ∆G U = −neU, where U is the electrode potential related to the standard hydrogen electrode. ∆G pH = k B TIn10 × pH is the correction for Gibbs free energy depending on the concentration of H + ions, and pH = 0 was used in this study. Moreover, the ∆G i of each reaction step of OER can be expressed as ∆G 1 = ∆G OH* , ∆G 2 =∆G O* −∆G OH* , ∆G 3 =∆G OOH* −∆G O* and ∆G 4 = 4.92 − ∆G OOH* . The overpotential for OER (η OER ) was calculated according to Nørskov's assumption [49]: where ∆G i (i = 1-4) and 1.23 represent the Gibbs free energy change for step (i) and the equilibrium potential.

Conclusions
Through the detailed DFT calculations, the electrocatalytic HER and OER performance of R-graphyne and TM@R-graphyne (TM = Fe, Ru, Os, Co, Rh, Ir, Ni, Pd and Pt) has been systematically investigated. The following interesting findings can be mainly obtained: (1) Compared with the inert graphene, 2D R-graphyne can exhibit certain HER electrocatalytic activity, which is mainly due to the anti-aromaticity of all its structural unit rings. Therefore, designing the 2D carbon allotrope completely composed of anti-aromatic structural units can be considered as an effective strategy to boost the HER catalytic activity of related carbon materials.
(2) Embedding the Group VIIIB atoms (i.e., Fe, Ru, Os, Co, Rh, Ir, Ni, Pd and Pt) can be considered as an effective approach to improve the HER catalytic activity of Rgraphyne, where the C-atoms near the TM-doping site can usually be used as highly active sites. In particular, Fe@R-graphyne, Os@R-graphyne, Rh@R-graphyne and Ir@R-graphyne can exhibit higher HER activity in comparison with other TM@R-graphynes due to the formation of more active sites. Closer distances between the relevant C atoms and the TM-doping site can generally endow them with the better HER catalytic activity. The improvement of HER performance can be due to the electron transfer process between the TM and its adjacent C atoms, which effectively regulates the electron density on the relevant C atoms, thus bringing the appropriate adsorption state of H*.
(3) Doping Co, Ni and Rh atoms in Group VIIIB can effectively enhance the OER catalytic performance of 2D R-graphyne. In particular, Ni@R-graphyne and Rh@R-graphyne can be identified as promising electrocatalysts, being capable of driving OER under very small overpotentials of 0.31 V and 0.48 V, respectively. ∆G O* can be considered to be a good descriptor for their OER activities. Further, we analyzed the reasons for the different OER activities of TM@R-graphyne systems from the perspective of molecular orbital theory. In addition, Ni@R-graphyne and Rh@R-graphyne are also expected to become bifunctional electrocatalysts for HER and OER.
Coupled with good conductivity and high structural stability, these 2D nanostructures based on R-graphyne can be considered as new and promising catalysts for HER or OER. All these intriguing findings can provide important insights for promoting the development of carbon-based electrocatalysts for water splitting.
Supplementary Materials: The following supporting information can be downloaded at https: //www.mdpi.com/article/10.3390/molecules28093888/s1, Figure S1: Three different C-C bonds are labeled, as well as two possible adsorption sites (S1 and S2) of TM atom; Figure S2: The optimized configurations of TM@R-graphyne (top view and side view), in which the doped TM atoms are Fe (a), Co (b), Ni (c), Ru (d), Rh (e), Pd (f), Os (g), Ir (h) and Pt (i), respectively; Figure S3: The total energy curve of ab initio molecular dynamics (AIMD) simulation for TM@R-graphyne at 500 K, where the inset is the configuration after 5000 fs; Figure S4: The density of states (DOS) of TM@Rgraphyne, where the Fermi energy was set to zero (black dash line); Figure S5: The Gibbs free energy diagrams for OER on Ru@R-graphyne and Ir@R-graphyne at an electrode potential of 0 V; Table S1: The optimized lattice constants (a = b) and the calculated binding energy (E b ) for the TM@R-graphyne; Table S2